Fair and Efficient Allocations under Subadditive Valuations

We study the problem of allocating a set of indivisible goods among agents with subadditive valuations in a fair and efficient manner. Envy-Freeness up to any good (EFX) is the most compelling notion of fairness in the context of indivisible goods. Although the existence of EFX is not known beyond the simple case of two agents with subadditive valuations, some good approximations of EFX are known to exist, namely $\tfrac{1}{2}$-EFX allocation and EFX allocations with bounded charity. Nash welfare (the geometric mean of agents' valuations) is one of the most commonly used measures of efficiency. In case of additive valuations, an allocation that maximizes Nash welfare also satisfies fairness properties like Envy-Free up to one good (EF1). Although there is substantial work on approximating Nash welfare when agents have additive valuations, very little is known when agents have subadditive valuations. In this paper, we design a polynomial-time algorithm that outputs an allocation that satisfies either of the two approximations of EFX as well as achieves an $\mathcal{O}(n)$ approximation to the Nash welfare. Our result also improves the current best-known approximation of $\mathcal{O}(n \log n)$ and $\mathcal{O}(m)$ to Nash welfare when agents have submodular and subadditive valuations, respectively. Furthermore, our technique also gives an $\mathcal{O}(n)$ approximation to a family of welfare measures, $p$-mean of valuations for $p\in (-\infty, 1]$, thereby also matching asymptotically the current best known approximation ratio for special cases like $p =-\infty$ while also retaining the fairness properties

Photoinduced Structural Dynamics across the Metal-Insulator Transition in Rare-earth Nickelates

Transition metal oxides (TMOs) display novel properties such as metal-insulator transitions, multiferroicity and superconductivity. TMOs displaying a metal-insulator transition (MIT) have emerged as potential building blocks for neuromorphic computing and oxide-based electronics. These materials are being considered as alternatives to semiconductor electronics to meet the increasing demands for faster, energy efficient computing requirements for internet of things, big data and cloud computing applications. In this dissertation, I focus on the rare-earth nickelates material system where the MIT temperature can be tuned to occur near room temperature.Laser-induced excitation drives the transition at ultrafast timescales and can be used in combination with time resolved x-ray diffraction to disentangle the contribution of competing spin, lattice and charge degrees of freedom. In this dissertation, photoinduced structural dynamics of rare-earth nickelate thin films, NdNiO3 and SmNiO3, grown on (001) oriented SrTiO3 were studied using time-resolved x-ray diffraction. The evolution of the (002) Bragg peak was tracked following laser excitation. The recovery pathways were found to be strongly dependent on laser fluence for NdNiO3 and distinct for the two rare-earth nickelates. The recovery of the (002) peak shifts was modeled using a one-dimensional thermal diffusion model which showed that the recovery processes are nonthermal at high fluences. For NdNiO3, the timescales for the recovery of the (002) peak shift were found to be closely related to the reported Ni magnetism recovery in NNO, potentially indicating magnetostructural coupling. Moreover, the evolution of integrated intensity and full width at half maximum points towards the presence of a structural phase separation during recovery.Time-resolved x-ray nanodiffraction was performed to measure the structural dynamics in NdNiO3 with picosecond temporal and nanometer spatial resolution. A spatially heterogenous photoinduced response was observed where localized areas with higher peak position and lower integrated intensity of the out of plane (220) Bragg peak underwent maximum expansion after laser excitation. A photoinduced double peak to single peak transformation was observed, indicating a localized insulator to metal transition. Presence of large strain gradients and propagation of strain waves at the speed of sound in NdNiO3 outwards from these localized areas was also observed. These results indicate the formation of photoinduced lateral thermal gradients in the film due to localized insulator – metal transitions which could be leveraged for the purpose of information transfer occuring at the speed of sound in neuromorphic computing based systems

On the Existence of Competitive Equilibrium with Chores

We study the chore division problem in the classic Arrow-Debreu exchange setting, where a set of agents want to divide their divisible chores (bads) to minimize their disutilities (costs). We assume that agents have linear disutility functions. Like the setting with goods, a division based on competitive equilibrium is regarded as one of the best mechanisms for bads. Equilibrium existence for goods has been extensively studied, resulting in a simple, polynomial-time verifiable, necessary and sufficient condition. However, dividing bads has not received a similar extensive study even though it is as relevant as dividing goods in day-to-day life. In this paper, we show that the problem of checking whether an equilibrium exists in chore division is NP-complete, which is in sharp contrast to the case of goods. Further, we derive a simple, polynomial-time verifiable, sufficient condition for existence. Our fixed-point formulation to show existence makes novel use of both Kakutani and Brouwer fixed-point theorems, the latter nested inside the former, to avoid the undefined demand issue specific to bads

Fast algorithms for rank-1 bimatrix games

The rank of a bimatrix game is the matrix rank of the sum of the two payoff matrices. This paper comprehensively analyzes games of rank one, and shows the following: (1) For a game of rank r, the set of its Nash equilibria is the intersection of a generically one-dimensional set of equilibria of parameterized games of rank r − 1 with a hyperplane. (2) One equilibrium of a rank-1 game can be found in polynomial time. (3) All equilibria of a rank-1 game can be found by following a piecewise linear path. In contrast, such a path-following method finds only one equilibrium of a bimatrix game. (4) The number of equilibria of a rank-1 game may be exponential. (5) There is a homeomorphism between the space of bimatrix games and their equilibrium correspondence that preserves rank. It is a variation of the homeomorphism used for the concept of strategic stability of an equilibrium component

Improving EFX Guarantees through Rainbow Cycle Number

We study the problem of fairly allocating a set of indivisible goods among $n$ agents with additive valuations. Envy-freeness up to any good (EFX) is arguably the most compelling fairness notion in this context. However, the existence of EFX allocations has not been settled and is one of the most important problems in fair division. Towards resolving this problem, many impressive results show the existence of its relaxations, e.g., the existence of $0.618$-EFX allocations, and the existence of EFX at most $n-1$ unallocated goods. The latter result was recently improved for three agents, in which the two unallocated goods are allocated through an involved procedure. Reducing the number of unallocated goods for arbitrary number of agents is a systematic way to settle the big question. In this paper, we develop a new approach, and show that for every $\varepsilon \in (0,1/2]$, there always exists a $(1-\varepsilon)$-EFX allocation with sublinear number of unallocated goods and high Nash welfare. For this, we reduce the EFX problem to a novel problem in extremal graph theory. We introduce the notion of rainbow cycle number $R(\cdot)$. For all $d \in \mathbb{N}$, $R(d)$ is the largest $k$ such that there exists a $k$-partite digraph $G =(\cup_{i \in [k]} V_i, E)$, in which 1) each part has at most $d$ vertices, i.e., $\lvert V_i \rvert \leq d$ for all $i \in [k]$, 2) for any two parts $V_i$ and $V_j$, each vertex in $V_i$ has an incoming edge from some vertex in $V_j$ and vice-versa, and 3) there exists no cycle in $G$ that contains at most one vertex from each part. We show that any upper bound on $R(d)$ directly translates to a sublinear bound on the number of unallocated goods. We establish a polynomial upper bound on $R(d)$, yielding our main result. Furthermore, our approach is constructive, which also gives a polynomial-time algorithm for finding such an allocation